GCSE METHODS IN MATHEMATICS (PILOT)

Transcription

1 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 1 For teaching from 2010 For awards from 2012 GCSE METHODS IN MATHEMATICS (PILOT) SPECIMEN ASSESSMENT MATERIALS

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3 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 3 Contents Page Question Papers 5 Unit 1 - Foundation Unit 1 - Higher Unit 2 - Foundation Unit 2 - Higher Mark Schemes and Assessment Grids 81 Unit 1 - Foundation Unit 1 - Higher Unit 2 - Foundation Unit 2 - Higher Summary Assessment Grids 97

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5 Candidate Name Centre Number Candidate Number GCSE PILOT (LINKED PAIR SCHEME) METHODS IN MATHEMATICS UNIT 1: METHODS (NON-CALCULATOR) SPECIMEN PAPER FOUNDATION TIER hours CALCULATORS ARE NOT TO BE USED FOR THIS PAPER INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces at the top of this page. Answer all the questions in the spaces provided. Take π as INFORMATION FOR CANDIDATES You should give details of your method of solution when appropriate. Unless stated, diagrams are not drawn to scale. Scale drawing solutions will not be acceptable where you are asked to calculate. The number of marks is given in brackets at the end of each question or part-question. You are reminded that assessment will take into account the quality of written communication (including mathematical communication) used in your answer to question 13. Question For Examiner s use Maximum Mark TOTAL MARK Mark Awarded CJ*MU1

6 2 Formula List Area of trapezium = 1 2 (a + b)h a h b Volume of prism = area of cross-section length crosssection length

7 3 Examiner 1. (a) (i) Write down in figures, the number seven thousand three hundred and six. [1] (ii) Write down in words, the number [1] (b) Using the following list of numbers Write down (i) two numbers that add up to 90, [1] (ii) two numbers that have a difference of 50, [1] (iii) the number which is 60 to the nearest 10, [1] (iv) the answer when 12 is multiplied by 6. [1] (c) Find all the factors of 21. [2] Turn over.

8 4 Examiner 2. (a) Use one of the following words to complete each of the following sentences. isosceles parallelogram rhombus equilateral hexagon rectangle square pentagon (i) Each of the angles of an... triangle is equal to sixty degrees. [1] (ii) A... has five sides. [1] (iii) All the sides of a... are equal and all the angles are right angles. [1] (b) In the diagram below, name the lines that are perpendicular to each other. B C A D The line... is perpendicular to the line... [1]

9 5 Examiner 3. Bethan throws a dice. On the probability scale below, mark the points A, B and C where A is the probability of throwing a 6, B C is the probability of throwing an even number, is the probability of having a total score of 1 when the dice is thrown twice. 0 1 [3] Turn over.

10 6 Examiner (a) Showing all your working, write 0 4,,, in order with the smallest first [3] (b) Mark Jones travels to London with his wife and their three children. While in London they visit a computer exhibition. The travel cost for each adult is 110. The cost of the computer exhibition is 10 per person. The total cost for the family of the travel and the visit to the computer exhibition is 390. Find the travel cost for each child. [4]

11 7 Examiner 5. (a) For the number machine below, INPUT Multiply by 4 Add 7 OUTPUT (i) find the value of the OUTPUT when the INPUT is 8, [1] (ii) find the value of the INPUT when the OUTPUT is 27. [1] (b) The following numbers have been produced using another number machine Complete the boxes for this number machine. INPUT OUTPUT [2] (c) Write down the next term in each of the following sequences 3, 10, 17, 24,... 80, 40, 20, 10,... [2] (d) Write in words, the rule for finding the next term in the following sequence. 4, 12, 36, 108,... [1] Turn over.

12 8 Examiner 6. (a) Find the size of x in the diagram below. 35 x Diagram not drawn to scale. [2] (b) D 20 A B C Diagram not drawn to scale. $ Find the size of ADC in the above diagram. [3]

13 9 Examiner 7. (a) Simplify each of the following. (i) 7x + 5x + 2x [1] (ii) 6y y 12 [1] (b) Given that a = 5b 6, find the value of a when b is 20. [2] (c) The coordinates of each of the points (1, 4), (2, 8) and (3, 12) satisfy a rule. The coordinates of the point (m, n) satisfy the same rule. Write down the rule that connects m and n. [2] Turn over.

14 10 Examiner 8. The diagram below shows four identical rectangles. y A B (12, 5) 0 x C Find the coordinates of the points A, B and C. The coordinates of A are (, ) The coordinates of B are (, ) The coordinates of C are (, ) [6]

15 11 Examiner 9. There are five green cards numbered 1, 3, 5, 7 and 9 respectively and four yellow cards numbered 2, 4, 6 and 8 respectively. In a game, a player chooses a green card and a yellow card at random. The score for the game is found by subtracting the smaller number on the two cards from the larger number on the two cards. For example, if the number on the green card is 1 and the number on the yellow card is 6, the player scores 5. (a) Complete the following table to show all the possible scores Yellow cards Green cards [2] A player wins a prize by getting a score of 5 or more. (b) Tony plays the game once. What is the probability that he wins a prize? [2] (c) 600 people each play the game once. Approximately how many would you expect to win a prize? [2] Turn over.

16 12 Examiner 10. In the end of year examinations in a school, 60 candidates sat History, 80 sat Spanish and 50 sat Film Studies. 20 candidates sat History and Spanish. 15 candidates sat Spanish and Film Studies. 25 candidates sat History and Film Studies. 12 candidates sat all three subjects. Draw a Venn diagram to show the above information and use it to find the total number of candidates who sat the end of year examinations. [6]

17 13 Examiner 11. Jim has one spin of the spinner shown below. BLUE RED GREEN YELLOW Diagram not drawn to scale. (a) The table below shows the probabilities of Jim obtaining YELLOW, GREEN or BLUE with one spin of the spinner. Complete the table by inserting the probability that Jim obtains RED with one spin of the spinner. Colour YELLOW GREEN BLUE RED Probability [2] (b) In a game, a player chooses two colours on the spinner and wins the game if either of the colours chosen is obtained with one spin of the spinner. Which two colours would you choose to have the best chance of winning? [1] (c) Find the probability of obtaining either GREEN or BLUE on the spinner with one spin of the spinner. [2] Turn over.

18 14 Examiner 12. A biased coin was tossed. The relative frequency of throwing a Head was calculated after a total of 20 throws, 40 throws, 60 throws, 80 throws and 100 throws. The results were plotted on the graph below. 1 Relative Frequency A B C D E Number of throws (a) Which one of the readings noted by the letters A, B, C D and E on the graph is likely to give the best estimate of the probability of throwing a Head with this coin? You must give a reason for your answer. [1]

19 15 Examiner (b) Using the graph, find how many times the coin (i) landed showing a head in the first 40 throws, (ii) landed showing a tail in the first 100 throws. [4] Turn over.

20 16 Examiner 13. (a) Complete the following table by placing a tick ( ) in any box where the given statement is true. Statement Square Parallelogram Trapezium The diagonals are equal in length Opposite angles are equal Only one pair of opposite sides are parallel The diagonals are lines of symmetry [3] (b) Explain why three lines of lengths 3cm, 5cm and 10cm cannot be used to form a triangle. [1] (c) You will be assessed on the quality of your written communication in this question. A six-sided polygon is to be drawn using a computer program. The designer has stated that three of the internal angles should be 140 each and the remaining three angles should all be acute angles. Write a report, with reasons, to the designer explaining whether or not this design is possible. [8]

21 Candidate Name Centre Number Candidate Number GCSE PILOT (LINKED PAIR SCHEME) METHODS IN MATHEMATICS UNIT 1: METHODS (NON-CALCULATOR) SPECIMEN PAPER HIGHER TIER 2 hours CALCULATORS ARE NOT TO BE USED FOR THIS PAPER INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces at the top of this page. Answer all the questions in the spaces provided. Take π as INFORMATION FOR CANDIDATES You should give details of your method of solution when appropriate. Unless stated, diagrams are not drawn to scale. Scale drawing solutions will not be acceptable where you are asked to calculate. The number of marks is given in brackets at the end of each question or part-question. You are reminded that assessment will take into account the quality of written communication (including mathematical communication) used in your answer to question 4. Question For Examiner s use Maximum Mark TOTAL MARK Mark Awarded CJ*MU1

22 2 Formula List a Area of trapezium = 1 2 (a + b)h h b Volume of prism = area of cross-section length crosssection length Volume of sphere = 4 πr 3 3 Surface area of sphere = 4πr 2 r Volume of cone = 1 πr 2 h 3 Curved surface area of cone = πrl l r h In any triangle ABC Sine rule a sin A b = = sin B c sin C C Cosine rule a 2 = b 2 + c 2 2bc cos A b a Area of triangle = 1 2 ab sin C A c B The Quadratic Equation The solutions of ax 2 + bx + c = 0 where a 0 are given by x = 2 b ± ( b 4ac) 2a

23 3 Examiner 1. (a) Use the formula below to find the value of c when d = 10 and e = 13. c = d(e + 6) 2 [3] (b) Make g the subject of the formula below. h = g + 2f [1] (c) Factorise 3k [1] (d) Simplify 4j + 10j 2(3j 7j). [2] Turn over.

24 4 Examiner 2. The diagram shows three parallel lines and another line that crosses the parallel lines. Find the angles marked a, b, c and d. a 70 b c d Diagram not drawn to scale. a =... b =... c =... d =... [4]

25 5 Examiner 3. (a) (i) Express 36 as a product of prime factors using index notation. [3] (ii) Find the highest common factor of 36 and 63. [2] (b) Write down the least common multiple of 18 and 30. [3] Turn over.

26 4. (a) y 6 Examiner P Q x The points P (3, 3) and Q (8, 3) are shown, on a centimetre square grid, on the above diagram. Another three points R, S and M are to be marked on this square grid. PQRS is a parallelogram. The point M is the mid-point of PR. Write down a possible set of coordinates for the points R, S and M. R (...,...) S (...,...) M (...,...) [4]

27 7 Examiner (c) Explain why three lines of lengths 3cm, 5cm and 10cm cannot be used to form a triangle. [1] (d) Two exterior angles of a triangle are 150 and 110. Calculate the size of the third exterior angle of the triangle. [3] (e) You will be assessed on the quality of your written communication in this question. A six sided polygon is to be drawn using a computer program. The designer has stated that three of the internal angles should be 140 each and the remaining three angles should all be acute angles. Write a report, with reasons, to the designer explaining whether or not this design is possible. [8] Turn over.

28 8 Examiner 5. Jim and Gwen are playing board games with spinners and dice. (a) Jim has one spin of the spinner shown below. RED BLUE GREEN YELLOW Diagram not drawn to scale. (i) The table below shows the probabilities of Jim obtaining YELLOW, GREEN or BLUE with one spin of the spinner. Complete the table by inserting the probability that Jim obtains RED with one spin of the spinner. Colour YELLOW GREEN BLUE RED Probability [2] (ii) In a game, a player chooses two colours on the spinner and wins the game if either of the colours chosen is obtained with one spin of the spinner. Which two colours would you choose to have the best chance of winning? [1] (iii) Find the probability of obtaining either GREEN or BLUE on the spinner. [2]

29 (b) 9 Examiner Gwen throws two fair dice, one coloured red and the other coloured white. She makes a note of the score on each dice. (i) Complete the following probability tree diagram to show the probabilities of events. Red dice White dice... Score of 4 or Score of 3... Score neither 4 nor 5... Score of 4 or 5... Score not 3... Score neither 4 nor 5 (ii) [3] Calculate the probability of Gwen not scoring 3 on the red dice and getting a score of 4 or 5 on the white dice. [2] (iii) Calculate the probability that Gwen gets a double three. [2] Turn over.

30 10 Examiner 6. (a) Copies of this shape are to be placed on the 16 by 16 grid below. The centre of the first copy of the shape is at (2, 5) as shown. y x The centre of the second and third copies of the shape are to be placed at (4, 8) and (6, 11). More shapes are placed on the grid so that they follow the same pattern. No part of the shape is drawn outside of the grid. By finding and writing down where the centre of the 20th copy of the shape would be placed, suggest the size of the smallest possible grid needed to allow the first 20 shapes to be drawn. [4]

31 11 Examiner (b) Write down the n th term of the sequence 7, 10, 13, 16, 19,... [2] (c) The diagrams show tile patterns. Each Pattern has some shaded tiles and some white tiles. Pattern 1 Pattern 2 Pattern 3 Pattern 4 Find the expression for the number of shaded tiles in Pattern n and an expression for the number of white tiles in Pattern n. [5] Turn over.

32 12 Examiner 7. Write down the value of each of the following, either as a whole number or as a fraction. (a) 9 0 (b) [1] [1] (c) 9 2 [1] (d) [1]

33 13 BLANK PAGE Turn over.

34 14 Examiner 8. Match the sketches of graphs below with the most appropriate equation. Make your choice of equation from the following list. y = 1 y = 1 y = y = x 2 y = x 3 x x x y = x y = x 3 y = x y = (x + 4) 2 y = (x 4) 2 y = x 2 y = x y = x y = x(x + 4) y = x(x 4) y 0 x Equation... y 0 x Equation...

35 15 Examiner y 0 x Equation... y 0 x Equation... y 0 x Equation... [4] Turn over.

36 16 Examiner 9. Find the length of the line marked w and the size of the angles marked x, y and z. You must give a reason for each of your answers. Diagrams are not drawn to scale. 10 cm 8cm 12 cm w [3] x [2]

37 17 Examiner y O 72 [2] z [2] Turn over.

38 18 Examiner 10. Each of ten cards has one number printed on it. Four of the ten cards have even numbers and the other six have odd numbers. The ten printed numbers are all different. Two cards are selected at random. (a) Calculate the probability that the sum of the two numbers on the selected cards is even. [4] (b) State why the probability of the product of the two numbers on the selected cards being a square number is NOT necessarily zero. [1]

39 19 BLANK PAGE Turn over.

40 11. (a) Expand and simplify (3x + 2)(4x 5). 20 Examiner [2] (b) Factorise the expression 10t t + 3 and hence solve the equation 10t t + 3 = 0. [3] (c) Factorise the expression 49d [2] (d) Express the following as a single fraction in its simplest form. 3 x 4 5 4x + 7 [4]

41 21 Examiner (e) Express x 2 + 6x 7 in the form (x + a) 2 + b where a and b are values to be found. [2] (f) Prove that 2x 5 x x = 13x [4] Turn over.

42 22 Examiner 12. A box contains 2 strawberry yoghurts, 4 vanilla yoghurts and 6 cherry yoghurts. Three yoghurts are selected at random from the box. Calculate the probability that at least one of the selected yoghurts is a cherry yoghurt. [3]

43 Candidate Name Centre Number Candidate Number GCSE PILOT (LINKED PAIR SCHEME) METHODS IN MATHEMATICS UNIT 2: METHODS (CALCULATOR) SPECIMEN PAPER FOUNDATION TIER hours ADDITIONAL MATERIALS A calculator will be required for this paper. INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces at the top of this page. Answer all the questions in the spaces provided. Take π as 3 14 or use the π button on your calculator. INFORMATION FOR CANDIDATES You should give details of your method of solution when appropriate. Unless stated, diagrams are not drawn to scale. Scale drawing solutions will not be acceptable where you are asked to calculate. The number of marks is given in brackets at the end of each question or part-question. You are reminded that assessment will take into account the quality of written communication (including mathematical communication) used in your answer to question 12. Question For Examiner s use Maximum Mark TOTAL MARK Mark Awarded CJ*MU2

44 2 Formula List Area of trapezium = 1 2 (a + b)h a h b Volume of prism = area of cross-section length crosssection length

45 3 Examiner 1. (a) In the following list, draw a circle around each ratio that is the same as 1:5. [2] 3:6 4:12 3:21 15:3 4:16 5:25 3:8 6:9 20 :100 (b) Complete the following table that shows equivalent fractions, decimals and percentages. Fraction Decimal Percentage % [4] (c) Twins Charlie and Sam are preparing for their birthday party. They make up party bags and share out red, green and yellow pencils. Each bag contains 10% red pencils, 25% green pencils and the remaining pencils in each bag are yellow. Each bag contains 20 pencils. They make up 20 party bags. (i) How many pencils do they need altogether? [1] (ii) How many red pencils are there in each bag? [2] (iii) How many green and yellow pencils do they need for each bag? Number of green pencils in each bag =... Number of yellow pencils in each bag =... [2] Turn over.

46 4 Examiner 2. On the following diagrams draw lines to show, an arc of a circle a radius of a circle a tangent to a circle An arc of a circle has been drawn for you. [2] Arc Radius Tangent

47 5 BLANK PAGE Turn over.

48 6 Examiner 3. (a) Use the following diagrams to write down two pairs of congruent shapes. P Q R T S V U W X One pair of congruent shapes is... and... Another pair of congruent shapes is... and... [2]

49 7 Examiner (b) Use the following diagrams to write down two pairs of similar shapes. A B C D E F G I One pair of similar shapes is... and... Another pair of similar shapes is... and... [2] Turn over.

50 8 Examiner 4. (a) Draw all the lines of symmetry on the following figure. [1] (b) Complete the following diagram so that AB is a line of symmetry. [2] A B

51 9 Examiner (c) For each of the following shapes write in the table below the order of rotational symmetry. Shape A Shape B Shape C Order of rotational symmetry Shape A Shape B Shape C [2] Turn over.

52 10 Examiner 5. The diagram shows 12 small squares which form a rectangular wire grid. The length of the grid is 4a centimetres. 4a (a) Find the total length of wire required to make the grid. (b) Find the total area of the grid. [5]

53 11 Examiner 6. (a) Find the value of [1] (b) Find the value of ( ) [2] Turn over.

54 12 Examiner 7. (a) (i) Find 19% of 450. (ii) Find 3 of [4] (b) A small company makes wooden chairs. In 2008 the company made 550 chairs. (i) The company increased the number of chairs it made in 2009 by 28% of the 2008 figure. How many chairs did the company make in 2009? [3] (ii) Because of economic difficulties, the number of chairs made in 2010 is likely to decrease by 25% of the 2009 figure. How many chairs will the company make in 2010? [3]

55 13 Examiner 8. Wooden cubes are used to make the following solid. (a) How many cubes are needed to make the solid? [4] (b) The length of the side of each cube is 2cm. Find the volume of the solid. [2] Turn over.

56 14 Examiner 9. (a) Solve each of the following equations. (i) 8x + 4 = 7 [2] (ii) 5(x 3) = 50 [3] (b) 3x 2x + 5 The length of a rectangle is 2x + 5cm. The width of the rectangle is 3xcm. The perimeter of the rectangle is 65cm. Find the value of x. [4]

57 15 Examiner (c) The angles, measured in degrees, of a quadrilateral are x, 3x 9, 124 and 2x + 5. Find the value of x. [4] (d) (i) Solve the inequality 4y [2] (ii) Write down the smallest whole number that satisfies this inequality. [1] Turn over.

58 16 Examiner 10. (a) Enlarge the shape shown on the grid by a scale factor of 2 using A as the centre of enlargement. A ( ) 9 (b) Translate the triangle shown by. 3 y [3] x [1]

59 17 Examiner 11. A 8cm B 6cm C Diagram not drawn to scale. For the triangle ABC you are given that AC = 16cm. Calculate the perpendicular distance from B to AC. [6] Turn over.

60 18 Examiner 12. You will be assessed on the quality of your written communication in this question. A prism has a uniform cross-section in the shape of a right-angled triangle ABC. E D F A 3 5 cm 5 3 cm C 2 8 cm B Diagram not drawn to scale. Write a report to another pupil showing clearly how to calculate the volume of the prism. [8]

61 Candidate Name Centre Number Candidate Number GCSE PILOT (LINKED PAIR SCHEME) METHODS IN MATHEMATICS UNIT 2: METHODS (CALCULATOR) SPECIMEN PAPER HIGHER TIER 2 hours ADDITIONAL MATERIALS A calculator will be required for this paper. INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces at the top of this page. Answer all the questions in the spaces provided. Take π as 3 14 or use the π button on your calculator. INFORMATION FOR CANDIDATES You should give details of your method of solution when appropriate. Unless stated, diagrams are not drawn to scale. Scale drawing solutions will not be acceptable where you are asked to calculate. The number of marks is given in brackets at the end of each question or part-question. You are reminded that assessment will take into account the quality of written communication (including mathematical communication) used in your answer to question 6. Question For Examiner s use Maximum Mark TOTAL MARK Mark Awarded CJ*MU2

62 2 Formula List a Area of trapezium = 1 2 (a + b)h h b Volume of prism = area of cross-section length crosssection length Volume of sphere = 4 πr 3 3 Surface area of sphere = 4πr 2 r Volume of cone = 1 πr 2 h 3 Curved surface area of cone = πrl l r h In any triangle ABC Sine rule a sin A b = = sin B c sin C C Cosine rule a 2 = b 2 + c 2 2bc cos A b a Area of triangle = 1 2 ab sin C A c B The Quadratic Equation The solutions of ax 2 + bx + c = 0 where a 0 are given by x = 2 b ± ( b 4ac) 2a

63 3 Examiner 1. (a) Solve x = [1] (b) Write down the two solutions to x 2 = 81. [2] (c) Solve 4(3x + 1) = 40. [3] (d) (i) Solve the inequality 4y [2] (ii) Write down the smallest whole number that satisfies this inequality. [1] Turn over.

64 4 Examiner 2. (a) (i) Write 54 as a percentage of 90. [2] (ii) Increase 720 by 23%. [2] (b) Find the value of giving your answer correct to one decimal place [3] (c) Write in standard form. [1]

65 5 Examiner (d) Write correct to (i) two significant figures, [1] (ii) two significant figures in standard form. [1] (e) Find the sum of 3 of 784 and 2 of [3] Turn over.

66 6 Examiner 3. A 8cm B 6cm C Diagram not drawn to scale. (a) Calculate the area of triangle ABC, clearly stating the units of your answer. [4] (b) Working with the same triangle ABC you are now given that AC = 16 cm, calculate the perpendicular distance from B to AC. [3]

67 7 Examiner (c) A rectangle PABQ is joined onto the same triangle ABC. The length of PA is 20cm. A P B C Q Diagram not drawn to scale. The area of PABQ is 224cm 2. (i) Calculate the length of AB. [3] (ii) Calculate the perimeter of shape PACBQ. [2] Turn over.

68 8 Examiner 4. (a) Enlarge the shape shown on the grid by a scale factor of 2 using A as the centre for the enlargement. A (b) Reflect the triangle in the line x = 1. [3] y x [2]

69 ( ) (c) Translate the triangle shown by 9. 3 y 9 Examiner x (d) Indicate with the letter B on the diagram which one of the shapes shown may be obtained by rotating shape A through 90 clockwise about O. y [1] A O x [1] Turn over.

70 10 Examiner 5. (a) When a number is increased by 20% it becomes 240. What was the original number? [3] (b) The circumference of a circle is 8 π cm, find the radius of the circle. [2]

71 11 Examiner 6. You will be assessed on the quality of your written communication in this question. A prism has a uniform cross-section in the shape of a right-angled triangle ABC. D A 3 5 cm 5 3 cm C 2 8 cm B Diagram not drawn to scale. Write a report to another pupil showing clearly how to calculate the volume of the prism. [8] Turn over.

72 12 Examiner 7. Sara creates a new shade of paint by mixing black, red and white paint. She used x litres of black paint. She used five times as many litres of red paint than she used of black paint. She used 12 more litres of white paint than she used of red paint. Altogether she produced 672 litres of the new shade of paint. Form an equation in terms of x and solve it to find the number of litres of each of the three different colours of paint used. [5]

73 13 Examiner 8. A rectangle is shown in the diagram between two parallel lines. 28 Diagram not drawn to scale. The rectangle is of length 7 1cm and width 3 4 cm. Calculate the perpendicular distance between the parallel lines. [9] Turn over.

74 14 Examiner 9. A pebble is thrown vertically upwards with a speed of s metres per second. The pebble reaches a maximum height of h metres, before falling vertically downwards. It is known that h is directly proportional to the square of s. A pebble thrown with a speed of 10 metres per second reaches a maximum height of 5 metres. (a) Calculate the maximum height reached when a pebble is thrown with a speed of 5 metres per second. [5] (b) The pebble reaches a maximum height of 0 45 metres. Calculate the speed at which the pebble was thrown. [2]

75 15 Examiner 10. A square and a rectangle are such that the side of the square is equal in length to the shorter side of the rectangle. The sum of the areas of the square and the rectangle is 198 cm 2, and the sum of the perimeters is 80cm. Calculate, using an algebraic method, the dimensions of the rectangle. [7] Turn over.

76 16 Examiner 11. A length of plastic tube has a uniform circular cross-section. The radius of the circular hole in the centre is xcm. The thickness of the plastic is 3cm and the length of the plastic tube is 5xcm. Given that the volume of the plastic used to make the tube is 88π cm 3, find the radius of the circular hole correct to one decimal place. [8]

77 17 BLANK PAGE Turn over.

78 18 Examiner 12. The plan of a race track shows parallel sides and semicircular ends. x (a) The ratio of the diameter, x, to the length of one of the parallel sides, y, is 1 : 2. The area contained within the race track is 500m 2. Find the lengths marked x and y on the diagram. y [5]

79 19 Examiner (b) Find the dimensions, marked v and w on the diagram, of a similar race track which contains an area of 2000m 2. v w [5]

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81 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 81 METHODS IN MATHEMATICS (PILOT) UNIT 1 - FOUNDATION TIER MARK SCHEME Methods in Mathematics Specimen Paper Unit 1 Foundation Tier 1 (a) (i) 7,306 (ii) Twenty five million (b) (i) (ii) (iii) 59 (iv) 72 (c) 1, 3, 7, 21 2 (a) (i) equilateral (ii) pentagon (iii) square (b) AB AD 3 A at or near B at 0.5 C at 0 4(a) (b) Adults travel ( )220 Exhibition entry ( )50 Total child travel Child travel ( )40 5 (a) (i) 39 (ii) 5 (b) Add 2 Multiply by 2 Subtract 1 (c) 31 5 (d) Multiply by three Mark B B2 7 CAO CAO Comments CAO CAO for 2 or 3 correct factors with no incorrect factors CAO CAO CAO For a method which allows Comparison of the 3 terms For 2 correct CAO FT CAO CAO for 2 correct CAO CAO

82 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 82 Methods in Mathematics Specimen Paper Unit 1 Foundation Tier 6 (a) (b) Use of the properties of isos triangle in triangle ABD Use of the properties of isos triangle in triangle BCD Angle ADC = (a) (i) 14x (ii) 4y + 6 (b) a = = 94 (c) n = 4m (a) (b) or equivalent 20 Mark 5 B2 6 6 B2 B2 Or CAO CAO CAO CAO CAO for sight of 4m CAO CAO CAO -1 for each error Comments for 20 or for numerator 6 in a fraction less than 1 (c) Three circles with 12 correct 8 correct for History and Spanish 3 correct for Spanish and Film studies 13 correct for History and Film studies (any 2 correct) 142 candidates 11 (a)(i) Strategy, knowing that the probabilities add to (b) Yellow and Blue (c) = FT CAO CAO CAO FT FT E.g. Attempt to add all and subtract from 1, or noticing first 2 make 0.5 & working towards 0.5 FT their (a) if greater than either of these

83 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 83 Methods in Mathematics Specimen Paper Unit 1 Foundation Tier 12 (a) E e.g. More throws, Uses all the data (b) (i) = 30 (ii) = (a) / / / / (b) < 10 or equivalent in words / Mark E1 5 B3 Comments SC1 for sight of 0.36 or B2 for any 2 columns or rows correct for any 1 column or row correct For understanding of formation (c) Internal or external angle method, 4 180, or ( 3 140) = 300 Three angles < 90 0 each, Angles left sum must be < 270 Conclusion, not possible B2 QWC2 12 Must use of meaning of acute, so 300 or QWC2 Presents material in a coherent and logical manner, using acceptable mathematical form and with few, if any errors in spelling, punctuation and grammar. QWC 1 Presents materials in an organised manner, mainly using acceptable mathematical form, with some errors in spelling, punctuation and grammar. QWC 0 Evident weaknesses in organisation of material and errors in use of mathematical form and in spelling, punctuation and grammar.

84 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 84 ASSESSMENT GRID METHODS IN MATHEMATICS (PILOT) UNIT 1: METHODS (NON-CALCULATOR) FOUNDATION TIER AO1 (50% - 60%) Assessment Objectives (Raw Marks) AO2 (15% - 25%) AO3 (20% - 30%) Total Mark QWC Question Totals

85 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 85 GCSE METHODS IN MATHEMATICS (PILOT) UNIT 1 - HIGHER TIER MARK SCHEME 1.(a) = 2 = -35 (b) g = h 2f (c) 3(k + 4) (d) 4j - 10j 6j + 14j = 2j Methods in Mathematics Specimen Paper Unit 1 Higher Tier 2.(a) a = 70 0, b = 70 0, c = 110 0, d = (a)(i) Method of finding a prime factor 2, 2, 3, (ii) 9 (b) 90 4.(a) Parallelogram with R, S and M meeting criteria (Parallelogram needs to be PQRS) (c) < 10 or equivalent in words (d) Sight of 30 and 70 OR exterior total is ( ) and intention to subtract this from 180 OR 360 ( ) 100 (0) Mark 7 B4 4 B2 B3 8 B4 Comments FT incorrect evaluation of (-13+6) 10 CAO for each, FT from previous answers when logical FT their prime factors, provided not all unique. for 3 B2 for at least 2 correct multiples of 18 and 30, OR for at least 2 correct multiples of 18 or 30. OR equivalent alternative strategy OR B3 M mid-point PR with quadrilateral, coordinates R, S, M OR B2 Parallelogram, M not mid-point, coordinates R, S, M OR Coordinates R, S, M for their diagram For understanding of formation Intention rather than accurate notation CAO. SC1 for 80 if no other marks awarded (e) Internal or external angle method, 4 180, or ( 3 140) = 300 Three angles < 90 0 each, Angles left sum must be < 270 Conclusion, not possible B2 Or alternative leading to 720 or 360 Must use meaning of acute, so for 300 or QWC2 QWC2 Presents material in a coherent and logical manner, using acceptable mathematical form and with few, if any errors in spelling, punctuation and grammar. QWC 1 Presents materials in an organised manner, mainly using acceptable mathematical form, with some errors in spelling, punctuation and grammar. QWC 0 Evident weaknesses in organisation of material and errors in use of mathematical form and in spelling, punctuation and grammar. 16

86 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 86 Methods in Mathematics Specimen Paper Unit 1 Higher Tier 5.(a)(i) Strategy, knowing that the probabilities add to 1 (ii) Yellow and Blue (iii) = Mark Comments E.g. Attempt to add all and subtract from 1, or noticing first 2 make 0.5 & working towards 0.5 FT their (a) if greater than either of these (b) (i) 6 5 ( Red not 3) 2 1 or 6 3 (White 4 or 5) 4 2 or 6 3 (White not 4 or 5) (ii) 5 2 = 6 6 (iii) = or 18 6.(a) 20 th shape centre at ( 40,. ) Pattern in y coordinate, 5, 8, 11,.. is add 3 20 th shape centre at (., 62 ) Grid from 40.5 by 62.5 to 42 by 64 (b) 3n + 4 (c) Shaded: n OR (n + 2) 2 (4n + 3) White: 4n + 3 OR (n + 2) 2 (n 2 + 1) 7.(a) 1 (b) (+) 3 (c) (d) B2 B2 B Ignore incorrect cancelling throughout (b). Either branch, not contradicted Either branch, not contradicted. FT 1 P(White 4 or 5) FT their P(White 4 or 5) from the top branch Accept 3 OR 3n OR 3n + 2 FT their 20 th centre coordinates +1 or +2 for 3n + B0 for n + 3 n 2 + OR (n + 2) 2 (4n + 3) with missing brackets B2 for 4n + OR (n + 2) 2 (n 2 + 1) with missing brackets for 7, 11, 15, 19 with attempt to find n th term Accept In order: y=, y=-x 3, y=x 2 +4, y=-x 2, y=x(x-4) B4 x = 8 w w = 15 (cm) Intersecting chords E1 x = 50 Alternate segment theorem E1 y = 36 Angle at the centre is twice the angle at the E1 circumference z = 55 Cyclic quadrilateral sum of opposite angles 180 E1 OR B3 4 correct, OR B2 3 correct, OR 2 correct Calculation alone without reason does not gain E1 9

87 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 87 Methods in Mathematics Specimen Paper Unit 1 Higher Tier 10.(a) Strategy, P(even, even) and P(odd, odd) or seen = or equivalent 90 (b) Explanation, e.g. Square number can be made from 2 different numbers multiplied together, or accept an example e.g. product of 2 and 8 give a square number 11.(a) 12x 2 + 8x 15x 10 = 12x 2 7x -10 (b) ( 5t +3 ) (2t + 1 ) 3 1 and 5 2 (c) (7d - 9) (7d + 9) (d) Numerator 3(4x + 7) 5(x 4) Denominator (x 4) ( 4x + 7) 7x + 41 ( x - 4)(4x + 7) (e) (x + 3) 2-16 (f) Attempt to use common denominator 6(2 x) + 5( x - 1) + 3(3x + 5) x + 5x x x + 10 = and 13 x Mark E1 5 B2 B2 A2 B2 A2 Comments FT from one error in the 4 terms for (5t 1) (2t 3) or split mid term and 1 st step factor F.T. for pair of brackets for (7d 9 ) (7d 9 ) FT one error to allow or for incorrect expansion of denominator SC1 for sight of 7x+41 if no other marks awarded for a=3 and for b=-16 for 1 slip or no conclusion Special case: x both sides 30 12x+5x-5+9x+15 26x + 10 Convincing = 2(13x+5) P(no cherry) P(no cherry) = (= = ) Or equivalent complete strategy, idea Seen alone not part of further probabilities. OR full alternative with correct values (= = ) CAO

88 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 88 ASSESSMENT GRID METHODS IN MATHEMATICS (PILOT) UNIT 1: METHODS (NON-CALCULATOR) HIGHER TIER AO1 (50% - 60%) Assessment Objectives (Raw Marks) AO2 (15% - 25%) AO3 (20% - 30%) Total Mark QWC Question Totals

89 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 89 GCSE METHODS IN MATHEMATICS (PILOT) UNIT 2 - FOUNDATION TIER MARK SCHEME Methods in Mathematics Specimen Paper Unit 2 Foundation Tier 1 (a) 5:25 20:100 (b) (0).5 75(%) (%) Mark B2 B2 Comments -1 for each error 3 or equivalent fraction 30% 10 (c) (i) (ii) or equivalent CAO 2 (iii) green 5 yellow Radius Tangent 3 (a) P S or S P U X or X U (b) A I or I A E F or F E 4 (a) 2 lines of symmetry drawn (b) Correct diagram (c) (a) 4a 4 + 3a 5 16a + 15a 31a(cm) (b) Area of 1 square = a 2 total area = 12a 2 (cm 2 ) 6 (a) 9.54 (b) B2 B2 5 5 B2 3 CAO CAO CAO Centre to circumference CAO -1 for each error for 2 correct Or any correct method FT one error Or 4a 3a CAO CAO for 7.6 or 10.24

90 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 90 Methods in Mathematics Specimen Paper Unit 2 Foundation Tier 19 7 (a) (i) (ii) 78 Mark B2 Comments CAO 2 25 for or (b) (i) M2 for CAO Chairs made in 2009 = 704 FT for 704 CAO (ii) their M2 for Chairs made in 2010 = (a) (= 50) (= 24) 5 2 2(= 20) Number of cubes = 94 (b) Volume of 1 cube (8 cm 3 ) Volume of block 752 (cm 3 ) 9 (a) (i) 8x = 3 3 x = CAO for FT from (i) FT OR CAO CAO CAO for 528 CAO 50 (ii) 5x 15 = 50 or x 3 = 5 5 x = 65 or x = x = 13 (b) 6x + 4x x + 10 = 65 10x = 55 x = 5.5(cm) (c) x + 3x x + 5 6x = 360 6x = 240 x = 40 (d) (i) 4y > 10 y > 2.5 (ii) 3 m1 m1 16 FT FT FT until 2 nd error CAO FT their inequality

91 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 91 Methods in Mathematics Specimen Paper Unit 2 Foundation Tier 10.(a) Enlargement scale factor 2 Correct position (b) Correct translation 11. Use of area = base height = 8 6 = 24 (cm 2 ) 16 B to AC = 24 B to AC = 8 24 = 3 (cm) 12. AB 2 = = AB = 4.5(cm) Area cross section = 2.8 AB Volume = area cross section (05 cm 3 ) or 22.1 (cm 3 ) Mark B2 4 6 QW2 8 Comments 3 lines correct, or consistent incorrect scale Maybe embedded in volume calculation FT their area x-section CAO QWC2 Presents material in a coherent and logical manner, using acceptable mathematical form and with few, if any errors in spelling, punctuation and grammar. QWC 1 Presents materials in an organised manner, mainly using acceptable mathematical form, with some errors in spelling, punctuation and grammar. QWC 0 Evident weaknesses in organisation of material and errors in use of mathematical form and in spelling, punctuation and grammar.

92 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 92 ASSESSMENT GRID METHODS IN MATHEMATICS (PILOT) UNIT 2: METHODS (CALCULATOR) FOUNDATION TIER AO1 (50% - 60%) Assessment Objectives (Raw Marks) AO2 (15% - 25%) AO3 (20% - 30%) Total Mark QWC Question Totals

93 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 93 Methods in Mathematics Specimen Paper Unit 2 Higher Tier 1. (a) x = 100 (b) 9 and -9 (c) 3x + 1 = 10 OR 12x + 4 = 40 3x = 9 OR 12x = 36 x = 3 (d) (i) 4y > 10 GCSE METHODS IN MATHEMATICS (PILOT) UNIT 2- HIGHER TIER MARK SCHEME Mark B2 for each solution In (c) FT until 2 nd error Comments 10 y > OR equivalent 4 (ii) (a) (i) = 60 (%) 23 (ii) or (b) B2 for or FT (c) CAO (d) (i) (ii) (e) (a) Use of area = base height = 8 6 = 24 cm 2 (b) 16 B to AC = 24 B to AC = 8 24 = 3 (cm) (c)(i) 20 AB = AB = 20 AB = 11.2 (cm) (ii) PQ = 73.2 (cm) 13 U1 12 FT one error FT their AB used as PQ from (i)

94 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 94 Methods in Mathematics Specimen Paper Unit 2 Higher Tier 4.(a) Enlargement scale factor 2 Correct position (b) Correct reflection in x = 1 (c) Correct translation (d) Bottom right shape indicated 5. (a) (b) 2πr = 8π r = or equivalent 120 = AB 2 = = AB = 4.5(cm) Area cross section = 2.8 AB Volume = area cross section (05 cm 3 ) or 22.1 (cm 3 ) 7. Sight of terms 5x and 5x + 12 Their expression of 3 terms = 672 x + 5x + 5x + 12 = 672 x = 60 (litres of black paint) 300 and 312 (litres) 8. Strategy, heights from 2 right-angled triangles h 1 =7.1sin28 or h 1 =7.1cos62 h 1 = 3.3(32 cm) Correct angle placement for second right angled triangle h 2 =3.4cos28 or h 2 =3.4sin62 h 2 = 3.0(02 cm) Shortest distance = 6.3 (cm) Mark B2 B2 7 m1 5 QWC2 8 5 M2 M2 9 Comments 3 lines correct, or consistent incorrect scale a reflection in y=1 or either axis, OR for drawing x=1 Accept any unambiguous indication 240 use of 120 Maybe embedded in volume calculation FT their area x-section CAO QWC2 Presents material in a coherent and logical manner, using acceptable mathematical form and with few, if any errors in spelling, punctuation and grammar. QWC 1 Presents materials in an organised manner, mainly using acceptable mathematical form, with some errors in spelling, punctuation and grammar. QWC 0 Evident weaknesses in organisation of material and errors in use of mathematical form and in spelling, punctuation and grammar. Accept x sign included Correct equation CAO FT their x for 5x and 5x+12 SC2 for 60, 300 and 312, no equation, OR SC1 for 60 litres of black paint, no equation h h for 1 = sin 28 or 1 = cos FT their 28 or 62 h h for 2 = cos 28 or 2 = sin FT their h 1 +h 2 if both B marks awarded

95 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 95 Methods in Mathematics Specimen Paper Unit 2 Higher Tier 9.(a) h s 2 5 = k k = 100 Mark FT non linear Comments 5 h = h = 1.25(m) or equivalent 100 (b) = s 2 (= 9) 5 s = 3(m/s) 10. x 2 + xy = 198 6x + 2y = 80 or 3x + y = 40 x 2 + x(40 3x) = 198 2x 2 40x = 0 or x 2 20x + 99 = 0 (x 9)(x 11) = 0 or equivalent x = 9 (or 11) Other length 13 (cm) 11. π (x + 3) 2 5x - πx 2 5x (x + 3) 2 = x 2 + 6x + 9 (maybe embedded in working) 5πx πx πx - 5πx 3 or equivalent Convincing 30πx πx 30x x 88 = ± 45 4(30)( 88) x = FT k 25 1 FT k Or alternative method, similar breakdown of stages FT for their equations CAO or negative of either quadratic Factorising their quadratic or formula method CAO FT their x or y value for shortest side logic OR (2x + 3) 2 or (2x + 6) 2 expanded correctly Or equivalent with π throughout FT their quadratic. Allow equivalent containing π. Allow 1 slip. OR trial & improvement, 1 correct trial 45 ± x = 60 x = 1.1(197 ) 12.(a) Use of y = 2x 2 2 πx Area = 2x + 4 x 2 π 2 + = x = ( OR x 2 = ) π x = (m) and 2x = y = (m) (b) Strategy, area ratio, or area equation 2 v Length scale factor 2 OR 2000 = vw + π 2 Realising 2v = w, maybe implied in ratio method v = (2x =) (m) w = (2y =) (m) 8 S1 S1 10 OR trials leading to correct < 0 and > 0 comparison Negative value not required for context

96 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 96 ASSESSMENT GRID METHODS IN MATHEMATICS (PILOT) UNIT 2: METHODS (CALCULATOR) HIGHER TIER AO1 (50% - 60%) Assessment Objectives (Raw Marks) AO2 (15% - 25%) AO3 (20% - 30%) Total Mark QWC Question Totals

97 GCSE METHODS IN MATHEMATICS (PILOT) Specimen Assessment Materials 97 SUMMARY ASSESSMENT GRIDS METHODS IN MATHEMATICS (PILOT) FOUNDATION TIER AO1 (50% - 60%) Assessment Objectives AO2 (15% - 25%) AO3 (20% - 30%) Total Marks Unit Mark % Mark % Mark % % 19 24% 18 23% % 17 21% 20 25% 80 Totals 86 54% 36 22% 38 24% 160 HIGHER TIER AO1 (50% - 60%) Assessment Objectives AO2 (15% - 25%) AO3 (20% - 30%) Total Marks Unit Mark % Mark % Mark % % 22 22% 20 20% % 25 25% 22 22% 100 Totals % % 42 21% 200 GCSE Methods in Mathematics (Pilot) SAMs/ED 20 November 2009